Problem: Vera studies the extinction of the bear population of Siberia over time. The following function gives the number of bears $t$ years since Vera started tracking it: $B(t)=2190\cdot e^{-0.3 t}$ What is the instantaneous rate of change of the number of bears after $2$ years? Choose 1 answer: Choose 1 answer: (Choice A) A $1202$ years per bear (Choice B) B $1202$ bears per year (Choice C) C $-361$ years per bear (Choice D) D $-361$ bears per year
Explanation: Understanding the problem The function that represents the instantaneous rate of change of $B(t)$ is its derivative, $B'(t)$. Therefore, the instantaneous rate of change of the number of bears after $2$ years is $B'(2)$. Let's find $B'(t)$ and evaluate it at $t=2$. Finding $B'(t)$ $B'(t)=-657\cdot e^{-0.3 t}$ Finding $B'(2)$ $\begin{aligned} B'(2)&=-657\cdot e^{-0.3(2)} \\\\ &=-657\cdot e^{-0.6} \\\\ &\approx -361 \end{aligned}$ Interpreting units $B(t)$ is the number of ${\text{bears}}$ after $t$ ${\text{years}}$. Therefore, we measure its rate of change in ${\text{bears}}$ per ${\text{year}}$. In conclusion, the instantaneous rate of change of the number of bears after $2$ years is $-361$ bears per year. The rate of change is negative because the number of bears is decreasing.